engleski

A parametric family of quartic Thue equations

In this paper we prove that the Diophantine equation x^4 - 4cx^3y + (6c+2)x^2y^2 + 4cxy^3 + y^4 = 1, where c > 2 is an integer, has only the trivial solutions (1,0), (0,1). Using the method of Tzanakis, we show that solving this quartic Thue equation reduces to solving the system of Pellian equations (2c+1)U^2 - 2cV^2 = 1, (c-2)U^2 - cZ^2 = -2, and we prove that all solutions of this system are given by (U,V,Z) = (1,1,1).

Thue equations; simultaneous Pellian equations; linear forms in logarithms

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